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1.
In this paper, we prove the existence and the uniqueness of global solution for the Cauchy problem for the generalized Boussinesq equation. Under some assumptions, we also show that the L∞ norm of small solution of the Cauchy problem for the generalized Boussinesq equation decays to zero as t tends to the infinity. 相似文献
2.
In this article, we investigate the Cauchy problem for the generalized double dispersion equation in n-dimensional space. We establish the decay estimates of solution to the corresponding linear equation. Under smallness condition on the initial data, we prove the global existence and asymptotic behaviour of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle. 相似文献
3.
Goro Akagi 《Journal of Differential Equations》2011,250(4):1850-1812
This paper addresses the analysis of dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semigroup approach, since solutions of the Cauchy problem for the equation might not be unique. Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn equation as well as a semilinear parabolic equation with a nonlinear term involving gradients. 相似文献
4.
The construction of analogues of the Cauchy kernel is crucial for the solution of Riemann–Hilbert problems on compact Riemann
surfaces. A formula for the Cauchy kernel can be given as an infinite sum over the elements of a Schottky group, and this
sum is often used for the explicit evaluation of the kernel. In this paper a new formula for a quasi-automorphic analogue
of the Cauchy kernel in terms of the Schottky–Klein prime function of the associated Schottky double is derived. This formula
opens the door to finding new ways to evaluate the analogue of the Cauchy kernel in cases where the infinite sum over a Schottky
group is not absolutely convergent. Application of this result to the solution of the Riemann–Hilbert problem with a discontinuous
coefficient for symmetric automorphic functions is discussed.
Received: March 10, 2007. Accepted: April 11, 2007. 相似文献
5.
Tomasz Klimsiak 《Stochastic Processes and their Applications》2012,122(1):134-169
We consider the Cauchy problem for a semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle, the problem has a unique solution and we provide its stochastic representation in terms of reflected backward stochastic differential equations. We also prove regularity properties and approximation results for solutions of the problem. 相似文献
6.
In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equation
utt - uxx - auxxtt + bux4 - duxxt = f(u)xx
are proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given. 相似文献
utt - uxx - auxxtt + bux4 - duxxt = f(u)xx
are proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given. 相似文献
7.
Vladimir G. Romanov 《Milan Journal of Mathematics》2006,74(1):357-385
We discuss here a method for investigating inverse problems for hyperbolic equations based on combining an asymptotic expansion
of solutions to the equations in a neighborhood of a characteristic surface, the connections between coefficients of the expansion
and the unknown coefficients of the equation and the estimates of a solution to the Cauchy problem in terms of the data on
a time-like surface. We give some applications of this method to a number of inverse problems. Stability results to the inverse
problems are given.
This work was partly supported by the Russian Foundation for Basic Research (Grant No. 05-01-00171).
Received: December 2005 相似文献
8.
A mollification method for ill-posed problems 总被引:3,自引:0,他引:3
Summary.
A mollification method for a class of ill-posed
problems is suggested. The idea of the method is very simple and
natural: if the data are given inexactly then we try to find a
sequence of ``mollification operators" which map the improper
data into well-posedness classes of the problem (mollify the
improper data). Within these mollified data our problem becomes
well-posed. And when these facts are in hand we try to obtain
error estimates and optimal or ``quasi-optimal" mollification
parameters. The
method is working not only for problems in Hilbert spaces, but
also for problems in Banach spaces. Applications of the
method to concrete problems, like numerical differentiation,
parabolic equations backwards in time, the Cauchy problem for the
Laplace equation,
one- and multidimensional non-characteristic Cauchy problems for
parabolic equations (in infinite or finite domains),... give us
very sharp stability estimates of H\"older continuous type. In
these cases the method is optimal in the sense that it gives the
same order of H\"older continuous dependence on the data as for
the regularized problems. Furthermore, the method may be
implemented numerically using fast Fourier transforms. For the
first time a uniform stability estimate of H\"older continuous
type of the solution of the heat equation backwards in time in the
space for all could
be established by our mollification method. A new simple sharp
pointwise estimate of H\"older type for the weak solution of a
non-characteristic Cauchy problem for parabolic equations in a
finite domain is established.
Received June 25, 1993 / Revised version received February
18, 1994 相似文献
9.
Z. Kamont 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(3):767-778
This paper deals with the Cauchy problem for nonlinear first order partial functional differential equations. The unknown function is the functional variable in the equation and the partial derivatives appear in a classical sense. A theorem on the local existence of a generalized solution is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions of this system is proved by using a method of successive approximations. A method of bicharacteristics and integral inequalities are applied. 相似文献
10.
In this paper, the existence and the uniqueness of the global strong solution and the global classical solution for the Cauchy problem of the multidimensional generalized IMBq equation are proved. The nonexistence of the global solution for the Cauchy problem of the generalized IMBq equation is discussed. 相似文献
11.
The authors investigate the problem of identifying the domainG of a harmonic functionu such that Cauchy data are given on a known portion of the boundary ofG, while a zero Dirichlet condition is specified on the remaining portion of the boundary, which is to be found. Under certain conditions on the domainG, it is shown that the problem reduces to identifying the coefficients of an elliptic equation which, in turn, is converted into the problem of minimizing a functional. Under certain conditions onG, it is shown that the solution, if it exists, is unique. An application is pointed out for the problem of designing a vessel shape that realizes a given plasma shape.This work was completed with a financial support from the National Basic Research in the Natural Sciences. 相似文献
12.
G. M. Henkin 《Journal of Fixed Point Theory and Applications》2007,1(2):239-291
Large time asymptotic structure for solutions of the Cauchy problem for a generalized Burgers equation is determined. In particular,
Gelfand’s question about location of viscous shock waves for such equations is answered. 相似文献
13.
Germán Fonseca Felipe Linares Gustavo Ponce 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2013
We study the initial value problem associated to the dispersion generalized Benjamin–Ono equation. Our aim is to establish persistence properties of the solution flow in weighted Sobolev spaces and to deduce from them some sharp unique continuation properties of solutions to this equation. In particular, we shall establish optimal decay rate for the solutions of this model. 相似文献
14.
In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the multidimensional generalized Boussinesq equation are obtained. Furthermore, the blow‐up of the solution for the Cauchy problem of the generalized Boussinesq equation is proved. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
15.
In this paper, we study the Cauchy problem of semilinear heat equations. By introducing a family of potential wells, we first prove the invariance of some sets and isolating solutions. Then we obtain a threshold result for the global existence and nonexistence of solutions. Finally we discuss the asymptotic behavior of the solution. 相似文献
16.
This paper is mainly concerned with the periodic Cauchy problem for a generalized two-component μ-Hunter-Saxton system with analytic initial data. The analyticity of its solutions is proved in both variables, globally in space and locally in time. The obtained result can be also applied to its special cases—the classical integrable two-component Hunter-Saxton system, the generalized μ-Hunter-Saxton equation and the classical Hunter-Saxton equation. 相似文献
17.
Jiaqing Pan 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(15):5069-5080
This work studies the large time behavior of free boundary and continuous dependence on nonlinearity for the Cauchy problem of a degenerate parabolic partial differential equation with absorption. Our objective is to give an explicit expression of speed of propagation of the solution and to show that the solution depends on the nonlinearity of the equation continuously. 相似文献
18.
In this paper, we consider the long-time behavior of small solutions of the Cauchy problem for a generalized Boussinesq equation. A scattering operator and the nonlinear scattering for small amplitude solutions of the Boussinesq equation are established under certain hypotheses. 相似文献
19.
The paper considers the Cauchy problem for linear partial differential equations of non-Kowalevskian type in the complex domain. It is shown that if the Cauchy data are entire functions of a suitable order, the problem has a formal solution which is multisummable. The precise bound of the admissible order of entire functions is described in terms of the Newton polygon of the equation. 相似文献
20.
Vladimir Varlamov 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(6):957-985
The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related
to the Korteweg-de Vries (KdV) and the Kadomtsev-Petviashvili models. KdV can be obtained from the equation in question when
the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented
in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived
for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy
transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained
for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation.
Received: November 23, 2004; revised: March 13, 2005
Research is supported by US Department of Defense, under grant No. DAAD19-03-1-0204 相似文献